1 Modèle de base des centrales thermiques¶

1.1 Planification journalière du parc thermique¶

Variables de décision : Pour une type de centrale $X \in [A,B,C]$, et une heure de la journée $t\in {1,\dots,24}$, on définit : $$ N_t^{(X)} = \text{Nombre de centrales } X \text{allumées à } t\text{ h, (ENTIER)} $$ $$ P_t^{(X)} = \text{Puissance totale produite par les centrales } X \text{ à } t\text{ h, (CONTINUE)} $$ Contraintes : $$ N_t^{(X)} P_{min}^{(X)} \leq P_t^{(X)} \leq N_t^{(X)} P_{max}^{(X)} \text{ , Contraintes sur la puissance totale de chaque centrale} $$ $$ 0 \leq N_t^{(X)} \leq N^{(X)} \text{ , Contraintes sur le nombre de centrales allumées possible} $$ $$ \forall t, \sum_X P_t^{(X)} = D_t \text{ , Contrainte équilibre offre-demande} $$ Objectif : $$ \text{Minimiser} \sum_X \sum_T P_t^{(X)} C_{MWh}^{(X)} $$

Set parameter TokenServer to value "dev.cma.mines-paristech.fr"
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 168 rows, 144 columns and 360 nonzeros
Model fingerprint: 0xf31da896
Variable types: 72 continuous, 72 integer (0 binary)
Coefficient statistics:
  Matrix range     [1e+00, 4e+03]
  Objective range  [1e+00, 3e+00]
  Bounds range     [5e+00, 1e+01]
  RHS range        [2e+04, 4e+04]
Found heuristic solution: objective 1235375.0000
Presolve removed 162 rows and 139 columns
Presolve time: 0.02s
Presolved: 6 rows, 5 columns, 14 nonzeros
Found heuristic solution: objective 881275.00000
Variable types: 2 continuous, 3 integer (0 binary)

Root relaxation: objective 8.694000e+05, 2 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 869400.000    0    1 881275.000 869400.000  1.35%     -    0s
H    0     0                    869400.00000 869400.000  0.00%     -    0s
     0     0 869400.000    0    1 869400.000 869400.000  0.00%     -    0s

Explored 1 nodes (2 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)

Solution count 3: 869400 881275 1.23538e+06 

Optimal solution found (tolerance 1.00e-04)
Best objective 8.694000000000e+05, best bound 8.694000000000e+05, gap 0.0000%
Coût : 869400.0
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 168 rows, 144 columns and 360 nonzeros
Model fingerprint: 0x635b27b9
Coefficient statistics:
  Matrix range     [1e+00, 4e+03]
  Objective range  [1e+00, 3e+00]
  Bounds range     [5e+00, 1e+01]
  RHS range        [2e+04, 4e+04]
Presolve removed 153 rows and 99 columns
Presolve time: 0.00s
Presolved: 15 rows, 45 columns, 45 nonzeros

Iteration    Objective       Primal Inf.    Dual Inf.      Time
       0    8.5266000e+05   1.743750e+04   0.000000e+00      0s
      15    8.6940000e+05   0.000000e+00   0.000000e+00      0s

Solved in 15 iterations and 0.01 seconds (0.00 work units)
Optimal objective  8.694000000e+05
Coût : 869400.0

2 Coût d'opération¶

2.1 Coût de fonctionnement¶

Pour ce problème on modifie la fonction objectif.
Objectif : $$ \text{Minimiser} \sum_X \sum_T (P_t^{(X)} - P_{min}^{(X)}N_t^{(X)}) C_{MWh}^{(X)} + N_t^{(X)}C_{base}^{(X)} $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 168 rows, 144 columns and 360 nonzeros
Model fingerprint: 0x7066cec8
Variable types: 72 continuous, 72 integer (0 binary)
Coefficient statistics:
  Matrix range     [1e+00, 4e+03]
  Objective range  [1e+00, 2e+03]
  Bounds range     [5e+00, 1e+01]
  RHS range        [2e+04, 4e+04]
Found heuristic solution: objective 1271875.0000
Presolve removed 162 rows and 139 columns
Presolve time: 0.03s
Presolved: 6 rows, 5 columns, 14 nonzeros
Found heuristic solution: objective 987500.00000
Variable types: 2 continuous, 3 integer (0 binary)
Found heuristic solution: objective 985950.00000

Root relaxation: objective 9.787500e+05, 2 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 978750.000    0    1 985950.000 978750.000  0.73%     -    0s
H    0     0                    978900.00000 978750.000  0.02%     -    0s
     0     0 978750.000    0    1 978900.000 978750.000  0.02%     -    0s

Explored 1 nodes (2 simplex iterations) in 0.05 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)

Solution count 4: 978900 985950 987500 1.27188e+06 

Optimal solution found (tolerance 1.00e-04)
Best objective 9.789000000000e+05, best bound 9.789000000000e+05, gap 0.0000%
Coût : 978900.0

2.2 Coût du démarrage¶

Pour ce problème on modifie les variables de décisions et on y adapte les contraintes.

Variables de décision : Pour une type de centrale $X \in [A,B,C]$, et une heure de la journée $t\in {1,\dots,24}$, on définit : $$ N_t^{(X)} = \text{Nombre de centrales } X \text{allumées à } t\text{ h, (ENTIER)} $$ $$ N_{start,t}^{(X)} = \text{Nombre de centrales } X \text{démarrées à } t\text{ h, (ENTIER)} $$ $$ P_t^{(X)} = \text{Puissance totale produite par les centrales } X \text{ à } t\text{ h, (CONTINUE)} $$ Contraintes (par convention $N_{-1}^{(X)}=0$): $$ N_t^{(X)} P_{min}^{(X)} \leq P_t^{(X)} \leq N_t^{(X)} P_{max}^{(X)} \text{ , Contraintes sur la puissance totale de chaque centrale} $$ $$ N_{start,t}^{(X)} \leq N_t^{(X)} \leq N_{start,t}^{(X)}+N_{t-1}^{(X)} \text{ , Contraintes sur le nombre de centrales allumées possible} $$ $$ 0 \leq N_{start,t}^{(X)} \leq N^{(X)}-N_{t-1}^{(X)} \text{ , Contraintes sur le nombre de centrales démarrable possible} $$ $$ \sum_X P_t^{(X)} = D_t \text{ , Contrainte équilibre offre-demande} $$ Objectif : $$ \text{Minimiser} \sum_X \sum_T (P_t^{(X)} - P_{min}^{(X)}N_t^{(X)}) C_{MWh}^{(X)} + N_t^{(X)}C_{base}^{(X)} + N_{start,t}^{(X)}C_{start}^{(X)} $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 312 rows, 216 columns and 714 nonzeros
Model fingerprint: 0xafd71f6e
Variable types: 72 continuous, 144 integer (0 binary)
Coefficient statistics:
  Matrix range     [1e+00, 4e+03]
  Objective range  [1e+00, 2e+03]
  Bounds range     [5e+00, 1e+01]
  RHS range        [5e+00, 4e+04]
Found heuristic solution: objective 1474375.0000
Presolve removed 108 rows and 30 columns
Presolve time: 0.00s
Presolved: 204 rows, 186 columns, 522 nonzeros
Variable types: 42 continuous, 144 integer (0 binary)
Found heuristic solution: objective 1378930.0000

Root relaxation: objective 1.011257e+06, 52 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 1011257.14    0   26 1378930.00 1011257.14  26.7%     -    0s
H    0     0                    1019660.0000 1011257.14  0.82%     -    0s
H    0     0                    1018090.0000 1011257.14  0.67%     -    0s
H    0     0                    1016340.0000 1011257.14  0.50%     -    0s
H    0     0                    1016100.0000 1011257.14  0.48%     -    0s
     0     0 1014233.33    0    4 1016100.00 1014233.33  0.18%     -    0s
H    0     0                    1015860.0000 1014233.33  0.16%     -    0s
H    0     0                    1014650.0000 1014233.33  0.04%     -    0s
H    0     0                    1014400.0000 1014233.33  0.02%     -    0s

Cutting planes:
  Gomory: 3
  MIR: 27
  StrongCG: 2

Explored 1 nodes (92 simplex iterations) in 0.04 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)

Solution count 9: 1.0144e+06 1.01465e+06 1.01586e+06 ... 1.47438e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 1.014400000000e+06, best bound 1.014400000000e+06, gap 0.0000%

3 Réserve de puissance¶

Pour intégrer la réserve de puissance on ajoute la contrainte : $$ \sum_X N_t^{(X)}P_{max}^{(X)}\geq D_t\times 1,15 $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 336 rows, 216 columns and 786 nonzeros
Model fingerprint: 0x11b5f8ee
Variable types: 72 continuous, 144 integer (0 binary)
Coefficient statistics:
  Matrix range     [1e+00, 4e+03]
  Objective range  [1e+00, 2e+03]
  Bounds range     [5e+00, 1e+01]
  RHS range        [5e+00, 5e+04]

MIP start from previous solve did not produce a new incumbent solution
MIP start from previous solve violates constraint Réserve_de_puissance_à_15 by 4500.000000000

Found heuristic solution: objective 1316565.0000
Presolve removed 108 rows and 30 columns
Presolve time: 0.00s
Presolved: 228 rows, 186 columns, 594 nonzeros
Found heuristic solution: objective 1229165.0000
Variable types: 42 continuous, 144 integer (0 binary)
Found heuristic solution: objective 1198245.0000

Root relaxation: objective 1.012257e+06, 59 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 1012257.14    0   26 1198245.00 1012257.14  15.5%     -    0s
H    0     0                    1020660.0000 1012257.14  0.82%     -    0s
H    0     0                    1017340.0000 1012257.14  0.50%     -    0s
H    0     0                    1017100.0000 1012257.14  0.48%     -    0s
     0     0 1015150.00    0   12 1017100.00 1015150.00  0.19%     -    0s
H    0     0                    1016860.0000 1015150.00  0.17%     -    0s
H    0     0                    1015725.0000 1015150.00  0.06%     -    0s
H    0     0                    1015150.0000 1015150.00  0.00%     -    0s

Cutting planes:
  Gomory: 1
  MIR: 25

Explored 1 nodes (94 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)

Solution count 9: 1.01515e+06 1.01572e+06 1.01686e+06 ... 1.31657e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 1.015150000000e+06, best bound 1.015150000000e+06, gap 0.0000%

4 Planification cyclique¶

Pour ce problème on modifie la convention $N_{-1}^{(X)}=0$ par $N_{-1}^{(X)}=N_{23}^{(X)}$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 336 rows, 216 columns and 792 nonzeros
Model fingerprint: 0xfaf92924
Variable types: 72 continuous, 144 integer (0 binary)
Coefficient statistics:
  Matrix range     [1e+00, 4e+03]
  Objective range  [1e+00, 2e+03]
  Bounds range     [5e+00, 1e+01]
  RHS range        [5e+00, 5e+04]
Found heuristic solution: objective 1313700.0000
Presolve removed 105 rows and 27 columns
Presolve time: 0.00s
Presolved: 231 rows, 189 columns, 603 nonzeros
Found heuristic solution: objective 1225165.0000
Variable types: 42 continuous, 147 integer (0 binary)
Found heuristic solution: objective 1192195.0000

Root relaxation: objective 9.855143e+05, 54 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 985514.286    0   26 1192195.00 985514.286  17.3%     -    0s
H    0     0                    993725.00000 985514.286  0.83%     -    0s
H    0     0                    990405.00000 985514.286  0.49%     -    0s
H    0     0                    990165.00000 985514.286  0.47%     -    0s
H    0     0                    989030.00000 988540.000  0.05%     -    0s
     0     0 988540.000    0    3 989030.000 988540.000  0.05%     -    0s
H    0     0                    988540.00000 988540.000  0.00%     -    0s
     0     0 988540.000    0    3 988540.000 988540.000  0.00%     -    0s

Cutting planes:
  MIR: 18

Explored 1 nodes (85 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)

Solution count 8: 988540 989030 990165 ... 1.3137e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 9.885400000000e+05, best bound 9.885400000000e+05, gap 0.0000%

5 Centrales hydroélectriques¶

5.1¶

On ajoute les variables de décisions suivantes:
pour $Y\in[9,14]$ (pour les centrales de 900 MW et 1400MW),
$$ H_t^{(Y)} \in \{0,1\} \text{ , vaut 1 si la centrale $Y$ fonctionne à } t \text{ h 0 sinon} $$ $$ H_{start,t}^{(Y)} \in \{0,1\} \text{ , vaut 1 si la centrale $Y$ démarre à } t \text{ h 0 sinon} $$ Avec les contraintes : $$ H_{start,t}^{(Y)} \leq 1 - H_{t-1}^{(Y)} \text{ , S'il y a un démarrage alors la centrale n'était pas allumée} $$ Autre option : $$ H_{t}^{(Y)} \leq H_{start,t}^{(Y)} + H_{t-1}^{(Y)} \text{ , Si la centrale fonctionne alors elle était allumée ou elle démarre} $$ $$ H_{start,t}^{(Y)} \leq H_{t}^{(Y)} \text{ , La centrale fonctionne lorsqu'elle est est démarrée} $$ $$ \sum_X P_t^{(X)} + \sum_Y N_t^{(Y)}P^{(Y)}= D_t \text{ , Contrainte équilibre offre-demande} $$ $$ \sum_X N_t^{(X)}P_{max}^{(X)} + \sum_Y P^{(Y)}\geq D_t\times 1,15 \text{ , marges de sécurité} $$ L'objectif mis à jour devient: $$ \text{Minimiser} \sum_X \sum_t (P_t^{(X)} - P_{min}^{(X)}N_t^{(X)}) C_{MWh}^{(X)} + N_t^{(X)}C_{base}^{(X)} + N_{start,t}^{(X)}C_{start}^{(X)} + \sum_Y \sum_t H_t^{(Y)}C_{base}^{(Y)} + H_t^{(Y)}C_{start}^{(Y)} $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 432 rows, 312 columns and 1080 nonzeros
Model fingerprint: 0x568b7ef7
Variable types: 72 continuous, 240 integer (96 binary)
Coefficient statistics:
  Matrix range     [1e+00, 4e+03]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+01]
  RHS range        [5e+00, 4e+04]
Found heuristic solution: objective 1447255.0000
Presolve removed 120 rows and 0 columns
Presolve time: 0.00s
Presolved: 312 rows, 312 columns, 840 nonzeros
Variable types: 72 continuous, 240 integer (96 binary)
Found heuristic solution: objective 1429430.0000

Root relaxation: objective 8.884163e+05, 90 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 888416.250    0   30 1429430.00 888416.250  37.8%     -    0s
H    0     0                    906941.25000 888416.250  2.04%     -    0s
H    0     0                    893135.00000 888416.250  0.53%     -    0s
H    0     0                    890510.00000 888416.250  0.24%     -    0s
H    0     0                    890260.00000 888474.286  0.20%     -    0s
     0     0 890176.667    0   22 890260.000 890176.667  0.01%     -    0s

Cutting planes:
  Gomory: 2
  MIR: 33

Explored 1 nodes (132 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)

Solution count 6: 890260 890510 893135 ... 1.44726e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 8.902600000000e+05, best bound 8.901766666667e+05, gap 0.0094%

5.2¶

On ajoute les variables de décision suivante : $$ S_t = \text{Puissance appelée par le pompage à l'instant }t $$ Puis la contrainte : $$ \sum_tS_t d^{(S)} = \sum_{t,Y} H_t^{(Y)}d^{(Y)} $$ Où $d^{(S)}$ représente la hauteur d'eau élevée par MWh et $d^{(Y)}$ la hauteur d'eau prélevée lorsque la centrale $Y$ fonctionne.
On met à jour les contraintes suivante: $$ \sum_X P_t^{(X)} + \sum_Y H_t^{(Y)}P^{(Y)}-S_t= D_t \text{ , Contrainte équilibre offre-demande} $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 433 rows, 336 columns and 1176 nonzeros
Model fingerprint: 0x2bb1f118
Variable types: 96 continuous, 240 integer (96 binary)
Coefficient statistics:
  Matrix range     [3e-04, 4e+03]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+01]
  RHS range        [5e+00, 4e+04]
Found heuristic solution: objective 1468770.0000
Presolve removed 120 rows and 0 columns
Presolve time: 0.00s
Presolved: 313 rows, 336 columns, 936 nonzeros
Variable types: 96 continuous, 240 integer (96 binary)
Found heuristic solution: objective 1428970.0000

Root relaxation: objective 9.850143e+05, 145 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 985014.286    0   26 1428970.00 985014.286  31.1%     -    0s
H    0     0                    994935.00000 985014.286  1.00%     -    0s
H    0     0                    991890.00000 985014.286  0.69%     -    0s
H    0     0                    990145.00000 985014.286  0.52%     -    0s
     0     0 985814.126    0   48 990145.000 985814.126  0.44%     -    0s
     0     0 985814.126    0   48 990145.000 985814.126  0.44%     -    0s
H    0     0                    988290.00000 985814.126  0.25%     -    0s
H    0     0                    988040.00000 985814.126  0.23%     -    0s
     0     2 985814.126    0   48 988040.000 985814.126  0.23%     -    0s
H    2     4                    988002.00000 986022.114  0.20%   4.0    0s
H    5     8                    986630.00000 986022.114  0.06%   8.8    0s
H   21    13                    986290.00000 986051.521  0.02%   8.0    0s

Cutting planes:
  Gomory: 5
  MIR: 24

Explored 45 nodes (439 simplex iterations) in 0.10 seconds (0.04 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 986290 986630 988002 ... 1.46877e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 9.862900000000e+05, best bound 9.862900000000e+05, gap 0.0000%

5.3 Paliers de fonctionnement¶

Les variables de décisions concernant l'hydroélectricité deviennent : $$ H_t^{(Y,n)} \in \{0,1\} = \text{La centrale hydroélectrique } Y \text{ fonctionne au palier } n \text{ à l'instant } t $$ $$ H_{start,t}^{(Y)} \in \{0,1\} = \text{ La centrale hydroélectrique $Y$ démarre à l'instant $t$} $$ Les contraintes suivantes sont modifiées :

$$ \text{Equilibre offre-demande : } \sum_X P_t^{(X)} + \sum_{Y,n} H_t^{(Y,n)}P^{(Y,n)}-S_t= D_t $$$$ \text{Réserve de puissance : } \sum_X N_t^{(X)}P_{max}^{(X)} + \sum_Y P^{(Y,n_{max})}\geq D_t\times 1,15 $$$$ \text{Niveau réservoir : } \sum_t S_t d^{(S)} = \sum_{t,Y,n} H_t^{(Y,n)}d^{(Y,n)} $$

Les contraintes sur les démarrages et les paliers deviennent : $$ \text{Un seul palier fonctionne à la fois , }\sum_n H_t^{(Y,n)} \leq 1 $$ $$ \text{ Si un palier fonctionne alors il y avait déjà un palier actif ou la centrale est démarrée , } H_t^{(Y,n)} \leq \sum_n H_{t-1}^{(Y,n)} + H_{start,t}^{(Y)} $$ Modification de la fonction objectif : $$ \text{Minimiser} \sum_X \sum_t (P_t^{(X)} - P_{min}^{(X)}N_t^{(X)}) C_{MWh}^{(X)} + N_t^{(X)}C_{base}^{(X)} + N_{start,t}^{(X)}C_{start}^{(X)} + \sum_{Y,n} \sum_t H_t^{(Y,n)}C_{base}^{(Y,n)} + \sum_Y H_{start,t}^{(Y)}C_{start}^{(Y)} $$ Où $P^{(Y,n)}$ est la puissance de la centrale $Y$ au palier $n$, $d^{(Y,n)}$ est l'abaissement sur une heure quand la centrale $Y$ fonctionne au palier $n$ et $C_{base}^{(Y,n)}$ est le coût du fonctionnement de la centrale $Y$ pendant une heure de fonctionnement au palier $n$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 577 rows, 480 columns and 3144 nonzeros
Model fingerprint: 0x0ee78a89
Variable types: 96 continuous, 384 integer (240 binary)
Coefficient statistics:
  Matrix range     [3e-04, 4e+03]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+01]
  RHS range        [1e+00, 4e+04]
Presolve removed 216 rows and 0 columns
Presolve time: 0.00s
Presolved: 361 rows, 480 columns, 1704 nonzeros
Variable types: 96 continuous, 384 integer (240 binary)
Found heuristic solution: objective 1114770.0000
Found heuristic solution: objective 1086990.0000
Found heuristic solution: objective 1026475.0000

Root relaxation: objective 9.850143e+05, 244 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 985014.286    0   26 1026475.00 985014.286  4.04%     -    0s
H    0     0                    994935.00000 985014.286  1.00%     -    0s
H    0     0                    990145.00000 985014.286  0.52%     -    0s
H    0     0                    987635.00000 985677.273  0.20%     -    0s
     0     0 985677.273    0   48 987635.000 985677.273  0.20%     -    0s
H    0     0                    987176.00000 985677.273  0.15%     -    0s
     0     0 985681.649    0   48 987176.000 985681.649  0.15%     -    0s
     0     0 985681.649    0   26 987176.000 985681.649  0.15%     -    0s
     0     0 985681.649    0   48 987176.000 985681.649  0.15%     -    0s
H    0     0                    987076.00000 985681.649  0.14%     -    0s
     0     0 985687.391    0   48 987076.000 985687.391  0.14%     -    0s
H    0     0                    986256.00000 985687.391  0.06%     -    0s
     0     0 985687.391    0   21 986256.000 985687.391  0.06%     -    0s
     0     0 985687.391    0   45 986256.000 985687.391  0.06%     -    0s
     0     0 985687.391    0   41 986256.000 985687.391  0.06%     -    0s
     0     0 985687.391    0   41 986256.000 985687.391  0.06%     -    0s
     0     2 985687.391    0   41 986256.000 985687.391  0.06%     -    0s
H   28     7                    986160.00000 985789.464  0.04%   9.4    0s

Cutting planes:
  Gomory: 2
  MIR: 22

Explored 50 nodes (1170 simplex iterations) in 0.21 seconds (0.07 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 986160 986256 987076 ... 1.11477e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 9.861600000000e+05, best bound 9.861380000000e+05, gap 0.0022%

5.4 Exclusion pompage et génération hydro¶

On introduit la variable de décision suivante : $$ N^{S}_t \in \{0,1\} = \text{La pompage est en fonctionnement} $$ On ajoute les contraintes suivante : $$ \text{Si une centrale hydroélectrique est activée alors le pompage est désactivé , } \frac{1}{\text{Nb centrale hydro}}\sum_{Y,n}H_t^{Y,n} \leq 1 - N_t^{(S)} $$ $$ \text{Si aucune centrale hydroélectrique est allumée alors il peut y avoir du pompage , } S_t \leq M \times N_t^{(S)} $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 625 rows, 504 columns and 3408 nonzeros
Model fingerprint: 0xa2175415
Variable types: 96 continuous, 408 integer (264 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+01]
  RHS range        [1e+00, 4e+04]
Presolve removed 240 rows and 0 columns
Presolve time: 0.01s
Presolved: 385 rows, 504 columns, 1800 nonzeros
Variable types: 96 continuous, 408 integer (264 binary)
Found heuristic solution: objective 1155935.0000
Found heuristic solution: objective 1086990.0000
Found heuristic solution: objective 1029100.0000

Root relaxation: objective 9.850143e+05, 210 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 985014.286    0   26 1029100.00 985014.286  4.28%     -    0s
H    0     0                    994935.00000 985014.286  1.00%     -    0s
H    0     0                    990145.00000 985014.286  0.52%     -    0s
H    0     0                    989730.00000 985677.273  0.41%     -    0s
     0     0 985677.273    0   66 989730.000 985677.273  0.41%     -    0s
     0     0 985692.849    0   72 989730.000 985692.849  0.41%     -    0s
H    0     0                    988290.00000 985692.849  0.26%     -    0s
     0     0 985697.389    0   72 988290.000 985697.389  0.26%     -    0s
H    0     0                    988040.00000 985697.389  0.24%     -    0s
     0     2 985697.389    0   72 988040.000 985697.389  0.24%     -    0s
H   35    40                    986855.00000 986007.287  0.09%  16.7    0s
H   75    56                    986771.00000 986031.356  0.07%  13.3    0s

Cutting planes:
  MIR: 54

Explored 335 nodes (4215 simplex iterations) in 0.21 seconds (0.14 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 986771 986771 986855 ... 1.08699e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 9.867710000000e+05, best bound 9.866744586729e+05, gap 0.0098%

6 Désagrégation¶

6.1 Individualisation des centrales¶

Variables de décision : Pour une type de centrale $X \in [A,B,C]$, $k$ l'identification de la $k$-ieme centrale, et une heure de la journée $t\in {1,\dots,24}$, on définit : $$ N_t^{(X,k)} \in\{0,1\} \text{La $k$-ieme centrale $X$ fonctionne à $t$} $$ $$ N_{start,t}^{(X,k)} \in\{0,1\} \text{La $k$-ieme centrale $X$ est démarrée à $t$} $$ $$ P_t^{(X,k)} = \text{Puissance de la $k$-ième centrale $X$ à l'instant $t$} $$ Contraintes : $$ N_t^{(X,k)} P_{min}^{(X)} \leq P_t^{(X,k)} \leq N_t^{(X,k)} P_{max}^{(X)} \text{ , Contraintes sur la puissance totale de chaque centrale} $$ $$ \text{Equilibre offre-demande : } \sum_X \sum_{k=1}^{N^{(X)}} P_t^{(X,k)} + \sum_{Y,n} H_t^{(Y,n)}P^{(Y,n)}-S_t= D_t $$ $$ \text{Réserve de puissance : } \sum_X \sum_{k=1}^{N^{(X)}} N_t^{(X,k)}P_{max}^{(X)} + \sum_Y P^{(Y,n_{max})}\geq D_t\times 1,15 $$ Objectif : $$ \text{Minimiser} \sum_X \sum_{k=1}^{N^{(X)}} \sum_t (P_t^{(X,k)} - P_{min}^{(X)}N_t^{(X,k)}) C_{MWh}^{(X)} + N_t^{(X,k)}C_{base}^{(X)} + N_{start,t}^{(X,k)}C_{start}^{(X)} + \sum_{Y,n} \sum_t H_t^{(Y,n)}C_{base}^{(Y,n)} + \sum_Y H_{start,t}^{(Y)}C_{start}^{(Y)} $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2281 rows, 2232 columns and 8448 nonzeros
Model fingerprint: 0x68b994eb
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 168 rows and 0 columns
Presolve time: 0.01s
Presolved: 2113 rows, 2232 columns, 6984 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1066070.0000
Found heuristic solution: objective 1049400.0000
Found heuristic solution: objective 1031950.0000

Root relaxation: objective 9.848205e+05, 1445 iterations, 0.02 seconds (0.01 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 984820.536    0   31 1031950.00 984820.536  4.57%     -    0s
H    0     0                    990145.00000 984820.536  0.54%     -    0s
H    0     0                    989980.00000 985527.783  0.45%     -    0s
     0     0 985527.783    0   75 989980.000 985527.783  0.45%     -    0s
H    0     0                    989730.00000 985527.783  0.42%     -    0s
     0     0 985527.783    0   75 989730.000 985527.783  0.42%     -    0s
     0     0 985550.204    0   98 989730.000 985550.204  0.42%     -    0s
     0     0 985550.204    0   98 989730.000 985550.204  0.42%     -    0s
H    0     0                    988837.00000 985550.204  0.33%     -    0s
     0     0 985680.000    0  100 988837.000 985680.000  0.32%     -    0s
H    0     0                    988697.00000 985680.000  0.31%     -    0s
     0     0 985680.000    0  105 988697.000 985680.000  0.31%     -    0s
     0     0 985680.000    0   94 988697.000 985680.000  0.31%     -    0s
     0     0 985680.000    0   73 988697.000 985680.000  0.31%     -    0s
     0     0 985680.000    0   73 988697.000 985680.000  0.31%     -    0s
     0     0 985680.000    0   73 988697.000 985680.000  0.31%     -    0s
H    0     0                    987354.00000 985680.000  0.17%     -    0s
     0     0 985680.000    0   28 987354.000 985680.000  0.17%     -    0s
     0     0 985680.000    0   68 987354.000 985680.000  0.17%     -    0s
     0     0 985680.000    0   80 987354.000 985680.000  0.17%     -    0s
     0     0 985681.190    0  102 987354.000 985681.190  0.17%     -    0s
     0     0 985681.190    0  102 987354.000 985681.190  0.17%     -    0s
     0     0 985681.548    0  101 987354.000 985681.548  0.17%     -    0s
     0     0 985681.548    0   80 987354.000 985681.548  0.17%     -    0s
H    0     0                    987114.00000 985681.548  0.15%     -    0s
     0     2 985681.548    0   79 987114.000 985681.548  0.15%     -    0s
H  289   189                    986827.00000 986050.750  0.08%  12.7    1s
H  343   220                    986818.00000 986165.714  0.07%  12.6    1s
H  431   223                    986817.99949 986165.714  0.07%  11.7    1s
* 6256   389              39    986775.00000 986657.391  0.01%   9.0    2s

Cutting planes:
  Cover: 15
  Implied bound: 3
  MIR: 32
  Flow cover: 43
  GUB cover: 3
  Inf proof: 2
  RLT: 6
  Relax-and-lift: 1

Explored 7129 nodes (66927 simplex iterations) in 3.11 seconds (3.66 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 986775 986818 986827 ... 989730

Optimal solution found (tolerance 1.00e-04)
Best objective 9.867750000000e+05, best bound 9.867288756614e+05, gap 0.0047%

6.3 Précision du profil de la demande¶

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2281 rows, 2232 columns and 8448 nonzeros
Model fingerprint: 0x65e31120
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 5e+04]
Presolve removed 168 rows and 0 columns
Presolve time: 0.01s
Presolved: 2113 rows, 2232 columns, 6984 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1070205.0000
Found heuristic solution: objective 1068650.0000

Root relaxation: objective 9.865919e+05, 1422 iterations, 0.01 seconds (0.01 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 986591.867    0   57 1068650.00 986591.867  7.68%     -    0s
H    0     0                    1001855.0000 986591.867  1.52%     -    0s
H    0     0                    999480.00000 986591.867  1.29%     -    0s
H    0     0                    992900.00000 986591.867  0.64%     -    0s
     0     0 986628.571    0  100 992900.000 986628.571  0.63%     -    0s
     0     0 986659.053    0   93 992900.000 986659.053  0.63%     -    0s
     0     0 986748.434    0   91 992900.000 986748.434  0.62%     -    0s
     0     0 986757.183    0  100 992900.000 986757.183  0.62%     -    0s
     0     0 986757.183    0  109 992900.000 986757.183  0.62%     -    0s
H    0     0                    990449.00000 986757.183  0.37%     -    0s
     0     0 986805.247    0  102 990449.000 986805.247  0.37%     -    0s
H    0     0                    989007.00000 986822.410  0.22%     -    0s
     0     0 986822.410    0   79 989007.000 986822.410  0.22%     -    0s
     0     0 986824.328    0   89 989007.000 986824.328  0.22%     -    0s
     0     0 986824.328    0  114 989007.000 986824.328  0.22%     -    0s
H    0     0                    988370.00000 986824.328  0.16%     -    0s
H    0     0                    988365.00000 986824.328  0.16%     -    0s
     0     0 986828.816    0  123 988365.000 986828.816  0.16%     -    0s
     0     0 986828.816    0   56 988365.000 986828.816  0.16%     -    0s
     0     0 986828.816    0  108 988365.000 986828.816  0.16%     -    0s
     0     0 986828.816    0   96 988365.000 986828.816  0.16%     -    0s
     0     0 986828.816    0  114 988365.000 986828.816  0.16%     -    0s
     0     0 986828.816    0  161 988365.000 986828.816  0.16%     -    0s
     0     0 986840.551    0  133 988365.000 986840.551  0.15%     -    0s
     0     0 986841.262    0  133 988365.000 986841.262  0.15%     -    0s
     0     0 986860.211    0  155 988365.000 986860.211  0.15%     -    0s
     0     0 986879.786    0  163 988365.000 986879.786  0.15%     -    0s
     0     0 986882.243    0  168 988365.000 986882.243  0.15%     -    0s
     0     0 986882.328    0  167 988365.000 986882.328  0.15%     -    0s
     0     0 986910.332    0  142 988365.000 986910.332  0.15%     -    0s
     0     0 986922.008    0  130 988365.000 986922.008  0.15%     -    0s
     0     0 986928.169    0  134 988365.000 986928.169  0.15%     -    0s
     0     0 986928.169    0  143 988365.000 986928.169  0.15%     -    0s
     0     0 986930.642    0  158 988365.000 986930.642  0.15%     -    0s
     0     0 986930.642    0  127 988365.000 986930.642  0.15%     -    0s
H    0     0                    988235.00000 986930.642  0.13%     -    0s
     0     2 986930.642    0  116 988235.000 986930.642  0.13%     -    0s
H   35    39                    988040.00000 987410.256  0.06%  14.9    0s
H   80    82                    988035.00000 987410.256  0.06%  12.2    0s
H   89    82                    988015.00000 987410.256  0.06%  11.6    0s
H  152   140                    987767.00000 987441.255  0.03%  10.3    0s
H  169   140                    987762.00000 987441.255  0.03%  10.0    0s
H  268   220                    987760.00000 987450.150  0.03%   8.7    1s
H  288   220                    987620.00000 987450.150  0.02%   8.9    1s
H  355   253                    987615.00000 987450.629  0.02%   9.7    1s

Cutting planes:
  Gomory: 3
  Implied bound: 5
  Projected implied bound: 1
  MIR: 11
  StrongCG: 1
  Flow cover: 12
  Relax-and-lift: 2

Explored 2469 nodes (24742 simplex iterations) in 2.67 seconds (1.73 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 987615 987615 987615 ... 988015

Optimal solution found (tolerance 1.00e-04)
Best objective 9.876150000000e+05, best bound 9.875199699458e+05, gap 0.0096%

6.4 Discrétisation à un pas de deux heures¶

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2281 rows, 2232 columns and 8448 nonzeros
Model fingerprint: 0x9893fba4
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 168 rows and 0 columns
Presolve time: 0.01s
Presolved: 2113 rows, 2232 columns, 6984 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1065810.0000
Found heuristic solution: objective 1049600.0000

Root relaxation: objective 9.892817e+05, 1484 iterations, 0.01 seconds (0.02 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 989281.715    0   54 1049600.00 989281.715  5.75%     -    0s
H    0     0                    1008655.0000 989281.715  1.92%     -    0s
H    0     0                    1003320.0000 989281.715  1.40%     -    0s
H    0     0                    991750.00000 989281.715  0.25%     -    0s
     0     0 989568.409    0   49 991750.000 989568.409  0.22%     -    0s
     0     0 989568.409    0   49 991750.000 989568.409  0.22%     -    0s
H    0     0                    990860.00000 989568.409  0.13%     -    0s
     0     0 989583.635    0   54 990860.000 989583.635  0.13%     -    0s
     0     0 989583.635    0   54 990860.000 989583.635  0.13%     -    0s
H    0     0                    990500.00000 989583.635  0.09%     -    0s
     0     0 989583.635    0   60 990500.000 989583.635  0.09%     -    0s
     0     0 989594.782    0   68 990500.000 989594.782  0.09%     -    0s
     0     0 989609.740    0   79 990500.000 989609.740  0.09%     -    0s
     0     0 989609.978    0   81 990500.000 989609.978  0.09%     -    0s
     0     0 989733.996    0   86 990500.000 989733.996  0.08%     -    0s
     0     0 989733.996    0   60 990500.000 989733.996  0.08%     -    0s
     0     0 989734.282    0   92 990500.000 989734.282  0.08%     -    0s
     0     0 989734.807    0   95 990500.000 989734.807  0.08%     -    0s
     0     0 989734.807    0   95 990500.000 989734.807  0.08%     -    0s
     0     0 989734.807    0  101 990500.000 989734.807  0.08%     -    0s
     0     0 989734.807    0   93 990500.000 989734.807  0.08%     -    0s
H    0     0                    990440.00000 989734.807  0.07%     -    0s
     0     0 989734.807    0   42 990440.000 989734.807  0.07%     -    0s
     0     0 989734.807    0   47 990440.000 989734.807  0.07%     -    0s
     0     0 989757.223    0   43 990440.000 989757.223  0.07%     -    0s
     0     0 989757.714    0   56 990440.000 989757.714  0.07%     -    0s
     0     0 989757.714    0   76 990440.000 989757.714  0.07%     -    0s
     0     0 989757.987    0   75 990440.000 989757.987  0.07%     -    0s
     0     0 989758.118    0   75 990440.000 989758.118  0.07%     -    0s
     0     0 989758.118    0   89 990440.000 989758.118  0.07%     -    0s
     0     0 989758.118    0   89 990440.000 989758.118  0.07%     -    0s
     0     0 989763.110    0   49 990440.000 989763.110  0.07%     -    0s
     0     0 989763.199    0   41 990440.000 989763.199  0.07%     -    0s
     0     0 989778.249    0   41 990440.000 989778.249  0.07%     -    0s
     0     0 989897.866    0   28 990440.000 989897.866  0.05%     -    0s
     0     0 989897.866    0   28 990440.000 989897.866  0.05%     -    0s
     0     2 989897.866    0   28 990440.000 989897.866  0.05%     -    0s
*  434   281              46    990430.00000 990122.540  0.03%   5.3    1s
H  979   626                    990420.00000 990122.540  0.03%   5.3    1s
H 1701   719                    990410.00000 990122.540  0.03%   6.1    1s

Cutting planes:
  Gomory: 9
  Lift-and-project: 1
  MIR: 79
  Flow cover: 53
  RLT: 2

Explored 26619 nodes (185253 simplex iterations) in 4.81 seconds (3.82 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 990410 990420 990430 ... 991750

Optimal solution found (tolerance 1.00e-04)
Best objective 9.904100000000e+05, best bound 9.903183333333e+05, gap 0.0093%

7 Disponibilité variable¶

7.1 Maintenance nocturne des centrales A¶

On ajoute la contrainte : $$ \forall t \in \{0,\dots,6\}\cup\{18,\dots,23\},\sum_{k=1}^{N^{(A)}}N_t^{(A,k)} \leq 10 $$

Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2294 rows, 2232 columns and 8604 nonzeros
Model fingerprint: 0xcd2d20ff
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 168 rows and 0 columns
Presolve time: 0.02s
Presolved: 2126 rows, 2232 columns, 7140 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1047065.0000
Found heuristic solution: objective 1045750.0000

Root relaxation: objective 1.003420e+06, 1703 iterations, 0.02 seconds (0.02 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 1003419.59    0   51 1045750.00 1003419.59  4.05%     -    0s
H    0     0                    1025155.0000 1003419.59  2.12%     -    0s
H    0     0                    1012530.0000 1003419.59  0.90%     -    0s
     0     0 1003700.84    0   91 1012530.00 1003700.84  0.87%     -    0s
H    0     0                    1011550.0000 1003700.84  0.78%     -    0s
H    0     0                    1011486.0000 1003700.84  0.77%     -    0s
     0     0 1003702.54    0  113 1011486.00 1003702.54  0.77%     -    0s
     0     0 1003702.54    0  120 1011486.00 1003702.54  0.77%     -    0s
     0     0 1003815.55    0   87 1011486.00 1003815.55  0.76%     -    0s
     0     0 1003906.43    0   59 1011486.00 1003906.43  0.75%     -    0s
H    0     0                    1008770.0000 1003906.43  0.48%     -    0s
     0     0 1004239.12    0  121 1008770.00 1004239.12  0.45%     -    0s
     0     0 1004239.12    0  122 1008770.00 1004239.12  0.45%     -    0s
     0     0 1004289.66    0   95 1008770.00 1004289.66  0.44%     -    0s
     0     0 1004289.66    0  111 1008770.00 1004289.66  0.44%     -    0s
     0     0 1004289.66    0   88 1008770.00 1004289.66  0.44%     -    0s
     0     0 1004289.66    0   99 1008770.00 1004289.66  0.44%     -    0s
     0     0 1004289.66    0   72 1008770.00 1004289.66  0.44%     -    0s
     0     0 1004289.66    0  107 1008770.00 1004289.66  0.44%     -    0s
     0     0 1004289.66    0   96 1008770.00 1004289.66  0.44%     -    0s
     0     0 1004289.66    0   75 1008770.00 1004289.66  0.44%     -    0s
H    0     0                    1006700.0000 1004289.66  0.24%     -    0s
     0     2 1004289.66    0   75 1006700.00 1004289.66  0.24%     -    0s
H   71    77                    1006622.0000 1004528.51  0.21%  15.2    0s
H  112   121                    1006305.0000 1004528.51  0.18%  12.6    0s
H  170   166                    1005559.0000 1004528.51  0.10%  10.8    0s
H  293   290                    1005405.0000 1004539.36  0.09%   9.2    1s
*  939   571              38    1005395.0000 1004565.24  0.08%   8.8    1s
H 1196   607                    1005350.0000 1004761.71  0.06%   5.0    4s
* 1313   608              45    1005300.0000 1004767.86  0.05%   5.2    4s
  4532  1299 1005184.17   41   31 1005300.00 1004783.47  0.05%   6.9    5s
 28672  3865 1005157.28   48   42 1005300.00 1005058.89  0.02%   8.5   10s

Cutting planes:
  Gomory: 36
  Implied bound: 1
  MIR: 31
  Flow cover: 22

Explored 35957 nodes (310944 simplex iterations) in 11.28 seconds (10.07 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 1.0053e+06 1.0053e+06 1.00535e+06 ... 1.00877e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 1.005300000000e+06, best bound 1.005206348214e+06, gap 0.0093%

8 Contraintes de rampe¶

8.1 Rampe croissante¶

$$ \forall t, \forall X , \forall k \in \{0,\dots,N^{(X)}\}, P_{t}^{(X,k)}-P_{t-1}^{(X,k)}\leq R_{montée}^{(X)} $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2929 rows, 2232 columns and 9744 nonzeros
Model fingerprint: 0xfe91f2fc
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 839 rows and 1269 columns
Presolve time: 0.02s
Presolved: 2090 rows, 963 columns, 6363 nonzeros
Variable types: 672 continuous, 291 integer (291 binary)

Root relaxation: objective 1.011158e+06, 1035 iterations, 0.01 seconds (0.01 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 1011158.36    0   42          - 1011158.36      -     -    0s
H    0     0                    1024447.0000 1011158.36  1.30%     -    0s
     0     0 1011766.70    0   37 1024447.00 1011766.70  1.24%     -    0s
     0     0 1012487.81    0   17 1024447.00 1012487.81  1.17%     -    0s
     0     0 1012487.81    0   17 1024447.00 1012487.81  1.17%     -    0s
H    0     0                    1019980.0000 1012487.81  0.73%     -    0s
     0     0 1012528.31    0   18 1019980.00 1012528.31  0.73%     -    0s
     0     0 1012528.31    0   18 1019980.00 1012528.31  0.73%     -    0s
     0     0 1012528.31    0   19 1019980.00 1012528.31  0.73%     -    0s
     0     0 1012528.31    0   18 1019980.00 1012528.31  0.73%     -    0s
     0     0 1012637.32    0   19 1019980.00 1012637.32  0.72%     -    0s
     0     0 1012637.32    0   17 1019980.00 1012637.32  0.72%     -    0s
H    0     0                    1017238.0000 1012637.32  0.45%     -    0s
     0     0 1012637.32    0   19 1017238.00 1012637.32  0.45%     -    0s
     0     0 1012637.32    0   17 1017238.00 1012637.32  0.45%     -    0s
H    0     0                    1017184.0000 1012637.32  0.45%     -    0s
     0     0 1012637.32    0   18 1017184.00 1012637.32  0.45%     -    0s
     0     0 1012637.32    0   18 1017184.00 1012637.32  0.45%     -    0s
     0     0 1012637.32    0   20 1017184.00 1012637.32  0.45%     -    0s
     0     0 1012637.32    0   20 1017184.00 1012637.32  0.45%     -    0s
H    0     0                    1016994.0000 1012637.32  0.43%     -    0s
     0     0 1012637.32    0   17 1016994.00 1012637.32  0.43%     -    0s
     0     2 1012637.32    0   17 1016994.00 1012637.32  0.43%     -    0s
H   27     3                    1016430.0000 1015144.08  0.13%  16.0    0s

Cutting planes:
  Gomory: 3
  MIR: 17
  Flow cover: 3
  RLT: 7

Explored 49 nodes (2033 simplex iterations) in 0.52 seconds (0.28 work units)
Thread count was 8 (of 8 available processors)

Solution count 7: 1.01643e+06 1.01643e+06 1.01699e+06 ... 1.02445e+06

Optimal solution found (tolerance 1.00e-04)
Best objective 1.016430000000e+06, best bound 1.016409639706e+06, gap 0.0020%

8.2 Rampe démarrage¶

$$ \forall t, \forall X , \forall k \in \{0,\dots,N^{(X)}\}, P_{t}^{(X,k)}-P_{t-1}^{(X,k)}\leq R_{montée}^{(X)} + N_{start,t}{(X,k)}(R_{démarrage}^{(X)}-R_{montée}^{(X)}) $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2929 rows, 2232 columns and 10392 nonzeros
Model fingerprint: 0x7fd3415b
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 168 rows and 0 columns
Presolve time: 0.02s
Presolved: 2761 rows, 2232 columns, 9576 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1143740.0000

Root relaxation: objective 9.865633e+05, 1541 iterations, 0.02 seconds (0.02 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 986563.290    0  133 1143740.00 986563.290  13.7%     -    0s
H    0     0                    1052540.0000 986563.290  6.27%     -    0s
H    0     0                    999785.00000 986563.290  1.32%     -    0s
H    0     0                    994685.00000 987113.421  0.76%     -    0s
     0     0 987113.421    0  126 994685.000 987113.421  0.76%     -    0s
     0     0 987306.646    0  120 994685.000 987306.646  0.74%     -    0s
     0     0 987886.769    0  121 994685.000 987886.769  0.68%     -    0s
     0     0 987887.226    0  122 994685.000 987887.226  0.68%     -    0s
     0     0 988302.092    0  135 994685.000 988302.092  0.64%     -    0s
     0     0 988304.690    0  140 994685.000 988304.690  0.64%     -    0s
     0     0 988309.044    0  152 994685.000 988309.044  0.64%     -    0s
     0     0 988309.044    0  152 994685.000 988309.044  0.64%     -    0s
     0     0 988336.401    0  113 994685.000 988336.401  0.64%     -    0s
     0     0 988343.477    0  123 994685.000 988343.477  0.64%     -    0s
     0     0 988345.222    0  123 994685.000 988345.222  0.64%     -    0s
H    0     0                    994500.00000 988351.317  0.62%     -    0s
     0     0 988351.317    0  124 994500.000 988351.317  0.62%     -    0s
     0     0 988355.092    0  135 994500.000 988355.092  0.62%     -    0s
H    0     0                    992340.00000 988355.092  0.40%     -    0s
H    0     0                    992270.00000 988355.092  0.39%     -    0s
     0     0 988361.795    0  111 992270.000 988361.795  0.39%     -    0s
     0     0 988361.907    0  112 992270.000 988361.907  0.39%     -    0s
     0     0 988374.433    0  125 992270.000 988374.433  0.39%     -    0s
     0     0 988374.740    0  120 992270.000 988374.740  0.39%     -    0s
     0     0 988375.536    0  142 992270.000 988375.536  0.39%     -    0s
     0     0 988375.800    0  146 992270.000 988375.800  0.39%     -    0s
     0     0 988375.905    0  146 992270.000 988375.905  0.39%     -    0s
     0     0 988375.905    0   95 992270.000 988375.905  0.39%     -    0s
     0     2 988375.905    0   94 992270.000 988375.905  0.39%     -    0s
H   31    40                    991247.00000 988715.567  0.26%  32.8    0s
H   33    40                    990205.00000 988715.567  0.15%  36.4    0s
H   78    74                    989795.00000 988715.567  0.11%  24.3    1s
*  188   129              43    989790.00000 988751.375  0.10%  18.6    1s
H  396   240                    989718.00000 988871.073  0.09%  16.4    1s
H  407   240                    989660.00000 988871.073  0.08%  16.1    1s
H  417   240                    989575.00000 988871.073  0.07%  16.0    1s
* 1017   490              53    989570.00000 988906.114  0.07%  12.7    1s
H 1201   513                    989565.00000 988929.986  0.06%  15.3    3s
* 1601   580              50    989425.00000 988957.672  0.05%  13.9    3s
* 1841   574              56    989415.00000 989028.615  0.04%  13.2    3s
* 2626   660              55    989352.00000 989102.739  0.03%  11.6    4s
* 2910   760              58    989350.00000 989113.997  0.02%  11.1    4s
  5902  1694     cutoff   53      989350.000 989181.643  0.02%   8.7    5s

Cutting planes:
  Gomory: 28
  Implied bound: 33
  MIR: 58
  Flow cover: 37
  Network: 1
  RLT: 8
  Relax-and-lift: 7

Explored 16360 nodes (126707 simplex iterations) in 7.60 seconds (6.30 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 989350 989352 989415 ... 989790

Optimal solution found (tolerance 1.00e-04)
Best objective 9.893500000000e+05, best bound 9.892542062638e+05, gap 0.0097%

8.3 Rampe décroissante et d'arrêt¶

$$ \forall t, \forall X , \forall k \in \{0,\dots,N^{(X)}\}, P_{t}^{(X,k)}-P_{t-1}^{(X,k)}\geq - R^{(X)}_{descente} $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 3577 rows, 2232 columns and 13632 nonzeros
Model fingerprint: 0xf96e20e4
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 168 rows and 0 columns
Presolve time: 0.02s
Presolved: 3409 rows, 2232 columns, 12168 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)

Root relaxation: objective 9.870197e+05, 1824 iterations, 0.03 seconds (0.03 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 987019.698    0  120          - 987019.698      -     -    0s
H    0     0                    1006722.0000 987019.698  1.96%     -    0s
H    0     0                    997562.00000 987019.698  1.06%     -    0s
H    0     0                    993270.00000 987019.698  0.63%     -    0s
     0     0 987417.844    0  141 993270.000 987417.844  0.59%     -    0s
     0     0 987539.149    0  142 993270.000 987539.149  0.58%     -    0s
     0     0 988614.725    0  163 993270.000 988614.725  0.47%     -    0s
     0     0 988875.471    0  172 993270.000 988875.471  0.44%     -    0s
     0     0 988892.108    0  134 993270.000 988892.108  0.44%     -    0s
     0     0 988893.176    0  140 993270.000 988893.176  0.44%     -    0s
     0     0 988893.176    0  140 993270.000 988893.176  0.44%     -    0s
     0     0 989269.307    0  197 993270.000 989269.307  0.40%     -    0s
     0     0 989274.817    0  173 993270.000 989274.817  0.40%     -    0s
     0     0 989275.150    0  180 993270.000 989275.150  0.40%     -    0s
     0     0 989284.627    0  194 993270.000 989284.627  0.40%     -    0s
     0     0 989285.754    0  206 993270.000 989285.754  0.40%     -    0s
     0     0 989287.369    0  201 993270.000 989287.369  0.40%     -    0s
     0     0 989290.155    0  231 993270.000 989290.155  0.40%     -    0s
     0     0 989291.326    0  229 993270.000 989291.326  0.40%     -    0s
     0     0 989292.122    0  211 993270.000 989292.122  0.40%     -    0s
     0     0 989292.188    0  212 993270.000 989292.188  0.40%     -    0s
     0     0 989296.378    0  242 993270.000 989296.378  0.40%     -    0s
     0     0 989297.973    0  255 993270.000 989297.973  0.40%     -    0s
     0     0 989298.240    0  280 993270.000 989298.240  0.40%     -    0s
     0     0 989302.239    0  277 993270.000 989302.239  0.40%     -    0s
     0     0 989302.639    0  234 993270.000 989302.639  0.40%     -    0s
     0     0 989302.789    0  272 993270.000 989302.789  0.40%     -    0s
     0     0 989302.789    0  256 993270.000 989302.789  0.40%     -    0s
H    0     0                    993125.00000 989302.789  0.38%     -    0s
     0     2 989302.789    0  243 993125.000 989302.789  0.38%     -    1s
H  161   100                    992915.00000 989305.457  0.36%  39.2    1s
H  180   115                    992805.00000 989308.155  0.35%  41.1    1s
H  290   169                    992340.00000 989308.524  0.31%  39.9    1s
H  301   183                    991085.00000 989309.688  0.18%  39.4    2s
H  332   191                    991070.00000 989312.678  0.18%  38.7    2s
H  335   191                    991030.00000 989312.678  0.17%  39.1    2s
*  616   301              70    990980.00000 989401.678  0.16%  31.1    2s
H 1078   510                    990955.00000 989469.138  0.15%  25.8    4s
  1080   512 990459.387   27  259 990955.000 989471.141  0.15%  25.8    5s
H 1272   580                    990945.00000 989573.094  0.14%  30.9    6s
H 1276   553                    990865.00000 989573.094  0.13%  30.8    6s
H 2990  1195                    990855.00000 990219.079  0.06%  26.6    8s
H 3333  1372                    990840.00000 990219.079  0.06%  25.8    8s
H 3366  1316                    990810.00000 990219.079  0.06%  25.7    8s
  4638  2098 990512.720   49   45 990810.000 990276.833  0.05%  25.6   10s
H 5197  2298                    990800.00000 990289.607  0.05%  25.8   10s
* 6821  3103              61    990780.00000 990319.343  0.05%  25.7   11s
  9576  4525 990411.262   40  146 990780.000 990342.643  0.04%  25.9   15s
 14637  5938 990539.690   43  256 990780.000 990384.653  0.04%  26.7   21s
 15031  6119 990496.749   62   52 990780.000 990384.653  0.04%  27.1   25s
H20428  7467                    990779.99985 990395.841  0.04%  24.3   28s
 24214  8548     cutoff   56      990780.000 990412.306  0.04%  23.4   30s
 34489  9625 990456.741   60  178 990780.000 990445.638  0.03%  23.3   35s
 43601 11997 990694.058   60   38 990780.000 990468.484  0.03%  23.9   40s
 44666 12242 990725.859   69   64 990780.000 990470.746  0.03%  23.9   47s
 49088 13265     cutoff   67      990780.000 990481.212  0.03%  24.2   50s
 56959 14829     cutoff   72      990780.000 990497.310  0.03%  24.4   55s
 65482 16194 990539.768   71   83 990780.000 990511.757  0.03%  24.6   60s
 73506 17311 990659.188   67   98 990780.000 990525.493  0.03%  24.7   65s
 81598 18172 990571.236   58   90 990780.000 990539.352  0.02%  24.8   70s
 89096 18802 990688.666   66   34 990780.000 990551.142  0.02%  24.8   75s
 97305 19225 990733.218   71   34 990780.000 990562.343  0.02%  25.0   80s
 104548 19604     cutoff   67      990780.000 990570.983  0.02%  25.0   85s
 111609 19603     cutoff   75      990780.000 990581.714  0.02%  25.1   90s
 118951 19261 990769.142   70  111 990780.000 990592.531  0.02%  25.1   95s
 123952 18823     cutoff   65      990780.000 990600.430  0.02%  25.1  100s
 131922 18217 990667.143   66   80 990780.000 990612.646  0.02%  25.1  105s
 138337 17488 990756.424   68   35 990780.000 990622.826  0.02%  25.0  110s
 144125 16623 990655.970   76  109 990780.000 990631.592  0.01%  25.0  115s
 150752 15761 990648.961   67   22 990780.000 990642.660  0.01%  24.9  120s
 157415 14928 990692.984   69   90 990780.000 990655.170  0.01%  24.9  125s
 163348 13906     cutoff   73      990780.000 990665.246  0.01%  24.8  130s
 170321 12903 990687.161   71   70 990780.000 990677.548  0.01%  24.6  135s

Cutting planes:
  Gomory: 4
  Implied bound: 41
  MIR: 1114
  Flow cover: 148
  Relax-and-lift: 1

Explored 173078 nodes (4253531 simplex iterations) in 136.97 seconds (178.34 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 990780 990780 990780 ... 990955

Optimal solution found (tolerance 1.00e-04)
Best objective 9.907800000000e+05, best bound 9.906827289231e+05, gap 0.0098%

9 Prévention de l'usure¶

9.1 Durée minimale d'activitée¶

On a la table suivante :


![Alt text](table.png)
$$ \forall t,\forall t' \in \{1,\dots,8\}, \forall k, N_{start,t}^{(X,k)}\leq N_{t+t'}^{(X,k)} , \text{La centrale fonctionne au moins 8 heures après le démarrage} $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 8545 rows, 2232 columns and 23568 nonzeros
Model fingerprint: 0x23695c7b
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 789 rows and 0 columns
Presolve time: 0.07s
Presolved: 7756 rows, 2232 columns, 21342 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1125610.0000

Root relaxation: objective 9.882113e+05, 2064 iterations, 0.04 seconds (0.04 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 988211.330    0  159 1125610.00 988211.330  12.2%     -    0s
H    0     0                    1125004.0000 988211.330  12.2%     -    0s
H    0     0                    1124844.0000 988260.979  12.1%     -    0s
     0     0 989271.123    0  191 1124844.00 989271.123  12.1%     -    0s
     0     0 989386.545    0  193 1124844.00 989386.545  12.0%     -    0s
     0     0 990895.926    0  161 1124844.00 990895.926  11.9%     -    0s
     0     0 991137.821    0  189 1124844.00 991137.821  11.9%     -    0s
     0     0 991140.925    0  168 1124844.00 991140.925  11.9%     -    0s
     0     0 991675.199    0  151 1124844.00 991675.199  11.8%     -    0s
     0     0 991681.986    0  167 1124844.00 991681.986  11.8%     -    0s
     0     0 991681.986    0  183 1124844.00 991681.986  11.8%     -    0s
     0     0 991687.956    0  157 1124844.00 991687.956  11.8%     -    0s
H    0     0                    1123854.0000 991687.956  11.8%     -    0s
H    0     0                    1002725.0000 991721.598  1.10%     -    0s
     0     0 991721.598    0  139 1002725.00 991721.598  1.10%     -    0s
     0     0 991721.941    0  146 1002725.00 991721.941  1.10%     -    0s
     0     0 991722.506    0  121 1002725.00 991722.506  1.10%     -    0s
     0     0 991722.506    0  184 1002725.00 991722.506  1.10%     -    0s
     0     0 991724.839    0  167 1002725.00 991724.839  1.10%     -    0s
H    0     0                    1002550.0000 991724.839  1.08%     -    0s
     0     0 991730.492    0  184 1002550.00 991730.492  1.08%     -    0s
     0     0 991730.492    0  184 1002550.00 991730.492  1.08%     -    0s
     0     0 991732.122    0  169 1002550.00 991732.122  1.08%     -    1s
     0     0 991732.410    0  186 1002550.00 991732.410  1.08%     -    1s
     0     0 991732.410    0  210 1002550.00 991732.410  1.08%     -    1s
     0     0 991732.410    0  195 1002550.00 991732.410  1.08%     -    1s
     0     2 991732.410    0  195 1002550.00 991732.410  1.08%     -    1s
H   32    38                    1002275.0000 991734.343  1.05%  79.6    1s
H   99   103                    995825.00000 991734.343  0.41%  84.9    1s
H  100   103                    995405.00000 991734.343  0.37%  84.2    1s
H  107   110                    995190.00000 991734.343  0.35%  80.4    1s
H  131   122                    995020.00000 991734.343  0.33%  70.5    1s
H  139   126                    994935.00000 991736.299  0.32%  67.6    1s
H  142   126                    994725.00000 991736.299  0.30%  69.4    1s
H  172   151                    994686.00000 991736.299  0.30%  60.7    1s
H  179   151                    994649.00000 991736.299  0.29%  58.6    1s
H  205   167                    994295.00000 991736.299  0.26%  56.4    2s
H  250   195                    993980.00000 991736.299  0.23%  51.7    2s
H  434   213                    993975.00000 991738.444  0.23%  44.2    2s
H  440   214                    993965.00000 991738.444  0.22%  44.1    2s
* 1191   520              71    993889.00000 991955.089  0.19%  43.8    4s
H 1319   634                    993864.78689 992003.841  0.19%  42.0    4s
H 1320   628                    993855.60656 992003.841  0.19%  42.0    4s
H 1324   627                    993854.90323 992003.841  0.19%  41.9    4s
* 1325   598              84    993812.00000 992003.841  0.18%  41.9    4s
H 1398   592                    993810.00000 992014.838  0.18%  41.0    4s
  1399   588 992911.797   11  195 993810.000 992014.838  0.18%  41.0    5s
  1432   613 992024.257   15  190 993810.000 992024.257  0.18%  43.2   10s
  4228  1126 993419.505   31   84 993810.000 993298.190  0.05%  29.5   15s
  9645  1816 993718.259   52   19 993810.000 993498.621  0.03%  23.9   20s
 15322  2680 993716.541   62    3 993810.000 993619.286  0.02%  21.8   25s
 22535  3542     cutoff   48      993810.000 993660.000  0.02%  19.8   31s
 27263  2973 993699.000   43    4 993810.000 993690.850  0.01%  19.8   35s

Cutting planes:
  Gomory: 3
  Lift-and-project: 3
  Implied bound: 39
  Clique: 3
  MIR: 193
  StrongCG: 2
  Flow cover: 54
  Zero half: 2
  RLT: 8
  Relax-and-lift: 3

Explored 31048 nodes (597858 simplex iterations) in 37.28 seconds (50.42 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 993810 993810 993812 ... 994686

Optimal solution found (tolerance 1.00e-04)
Best objective 9.938100000000e+05, best bound 9.937144285714e+05, gap 0.0096%

9.2 Durée minimale d'arrêt¶

$$ \forall t,\forall t' \in \{1,\dots,8\}, \forall k, 1 - (N_{t-1}^{(X,k)}-N_{t}^{(X,k)})\geq N_{t+t'}^{(X,k)}, \text{La centrale est éteinte pendant 8 heures après arrêts} $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 13081 rows, 2232 columns and 36204 nonzeros
Model fingerprint: 0xb9b1994c
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
  Matrix range     [3e-04, 6e+04]
  Objective range  [1e+00, 2e+03]
  Bounds range     [1e+00, 1e+00]
  RHS range        [1e+00, 4e+04]
Presolve removed 1356 rows and 0 columns
Presolve time: 0.10s
Presolved: 11725 rows, 2232 columns, 33438 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)

Use crossover to convert LP symmetric solution to basic solution...
Extra simplex iterations after uncrush: 24

Root relaxation: objective 9.886341e+05, 647 iterations, 0.05 seconds (0.04 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 988634.126    0  157          - 988634.126      -     -    0s
     0     0 989023.535    0  180          - 989023.535      -     -    0s
     0     0 989055.255    0  164          - 989055.255      -     -    0s
     0     0 990551.330    0  171          - 990551.330      -     -    0s
     0     0 991373.775    0  164          - 991373.775      -     -    0s
     0     0 991374.390    0  167          - 991374.390      -     -    0s
     0     0 991670.808    0  147          - 991670.808      -     -    0s
     0     0 991675.326    0  156          - 991675.326      -     -    0s
     0     0 991676.496    0  188          - 991676.496      -     -    0s
     0     0 991677.001    0  184          - 991677.001      -     -    1s
     0     0 991716.134    0  193          - 991716.134      -     -    1s
     0     0 991736.808    0  221          - 991736.808      -     -    1s
     0     0 991737.544    0  230          - 991737.544      -     -    1s
     0     0 991744.334    0  227          - 991744.334      -     -    1s
     0     0 991745.433    0  234          - 991745.433      -     -    1s
     0     0 991751.361    0  228          - 991751.361      -     -    1s
     0     0 991751.916    0  243          - 991751.916      -     -    1s
     0     0 991756.829    0  241          - 991756.829      -     -    1s
     0     0 991759.393    0  261          - 991759.393      -     -    1s
     0     0 991761.125    0  246          - 991761.125      -     -    1s
     0     0 991762.178    0  263          - 991762.178      -     -    1s
     0     0 991762.354    0  244          - 991762.354      -     -    1s
     0     0 991769.273    0  246          - 991769.273      -     -    1s
     0     0 991769.468    0  258          - 991769.468      -     -    1s
     0     0 991804.075    0  234          - 991804.075      -     -    1s
     0     0 991804.789    0  227          - 991804.789      -     -    1s
     0     0 991805.409    0  227          - 991805.409      -     -    1s
     0     0 991805.431    0  238          - 991805.431      -     -    1s
H    0     0                    1001045.0000 991805.431  0.92%     -    1s
H    0     0                    1000905.0000 991805.454  0.91%     -    2s
     0     0 991805.454    0  232 1000905.00 991805.454  0.91%     -    2s
     0     0 991806.205    0  231 1000905.00 991806.205  0.91%     -    2s
     0     0 991806.364    0  229 1000905.00 991806.364  0.91%     -    2s
     0     0 991806.786    0  238 1000905.00 991806.786  0.91%     -    2s
     0     0 991806.898    0  235 1000905.00 991806.898  0.91%     -    2s
     0     0 991807.061    0  235 1000905.00 991807.061  0.91%     -    2s
     0     0 991807.061    0  235 1000905.00 991807.061  0.91%     -    2s
     0     2 991807.061    0  235 1000905.00 991807.061  0.91%     -    2s
H  164   126                    1000766.0000 991810.720  0.89%   166    4s
H  242   180                    1000710.0000 991810.720  0.89%   154    4s
H  252   180                    1000640.0000 991810.720  0.88%   149    4s
H  255   180                    1000575.0000 991810.720  0.88%   153    4s
   348   287 997213.162    9  197 1000575.00 991811.259  0.88%   135    5s
H  385   277                    999950.00000 991811.259  0.81%   130    5s
H  386   274                    999920.00000 991811.259  0.81%   131    5s
H  386   272                    999612.00000 991811.259  0.78%   131    5s
H  391   267                    999250.00000 991811.259  0.74%   130    5s
H  432   283                    999240.00000 991811.259  0.74%   121    5s
H  465   248                    998290.00000 991811.259  0.65%   115    5s
H  568   286                    997935.00000 991924.397  0.60%   117    6s
H  576   281                    997845.00000 991924.397  0.59%   119    6s
H 1011   535                    997710.00000 991966.592  0.58%   100    7s
H 1012   528                    997625.00000 991966.592  0.57%   100    7s
H 1016   517                    997429.00000 991966.592  0.55%   101    7s
H 1026   358                    996165.00000 991966.592  0.42%   100    8s
  1039   359 993961.042   61  238 996165.000 991966.592  0.42%  98.8   10s
  1078   386 infeasible   15      996165.000 992001.717  0.42%   101   16s
H 1137   397                    995630.00000 994108.351  0.15%   102   17s
H 1171   388                    995480.00000 994108.351  0.14%   100   17s
H 1175   369                    995440.00000 994108.351  0.13%   100   17s
H 1201   374                    995305.00000 994108.351  0.12%  98.8   17s
H 1203   357                    995135.00000 994108.351  0.10%  98.7   17s
H 1205   341                    995085.00000 994108.351  0.10%  98.6   17s
H 1208   326                    994990.00000 994108.351  0.09%  98.3   17s
H 1242   320                    994815.00000 994108.351  0.07%  96.3   18s
H 1275   307                    994737.00000 994108.351  0.06%  94.3   18s
H 1319   286                    994715.00000 994108.351  0.06%  91.8   18s
H 1378   258                    994710.00000 994240.900  0.05%  89.0   18s
H 1497   261                    994705.00000 994372.220  0.03%  84.3   19s
  1867   320 994619.631   40   41 994705.000 994434.142  0.03%  72.1   20s
* 2182   352              58    994697.00000 994444.397  0.03%  64.4   20s
  5024   519     cutoff   38      994697.000 994578.833  0.01%  39.0   25s

Cutting planes:
  Gomory: 3
  Implied bound: 42
  Clique: 5
  MIR: 164
  Flow cover: 103
  Zero half: 23
  RLT: 14
  Relax-and-lift: 2

Explored 5909 nodes (215369 simplex iterations) in 25.70 seconds (31.61 work units)
Thread count was 8 (of 8 available processors)

Solution count 10: 994697 994705 994710 ... 995305

Optimal solution found (tolerance 1.00e-04)
Best objective 9.946970000000e+05, best bound 9.946060555556e+05, gap 0.0091%